Scientific notation

Scientific notation, also known as standard form or as exponential notation, is a way of writing numbers that accommodates values too large or small to be conveniently written in standard decimal notation. Scientific notation has a number of useful properties and is often favored by scientists, mathematicians and engineers, who work with such numbers.

In scientific notation all numbers are written like this:

("a times ten to the power of b"), where the exponent b is an integer, and the coefficient a is any real number, called the significand or mantissa (though the term "mantissa" may cause confusion as it can also refer to the fractional part of the common logarithm). If the number is negative then a minus sign precedes a (as in ordinary decimal notation).

Normalized notation

Any given number can be written in the form a × 10^b in many ways; for example 350 can be written as 3.5×10² or 35×10¹ or 350×10⁰.

In normalized scientific notation, the exponent b is chosen such that the absolute value of a remains at least one but less than ten (1 ≤ |a| < style="white-space: nowrap;">3.5×10². This form allows easy comparison of two numbers of the same sign in a, as the exponent b gives the number's order of magnitude. In normalized notation the exponent b is negative for a number with absolute value between 0 and 1 (e.g., minus one half is −5×10⁻¹). The 10 and exponent are usually omitted when the exponent is 0. Note that 0 itself cannot be written in normalised scientific notation; the exponent does not matter and the mantissa must be less than 1.

In many fields, scientific notation is normalized in this way, except during intermediate calculations or when an unnormalised form, such as engineering notation, is desired. (Normalized) scientific notation is often called exponential notation — although the latter term is more general and also applies when a is not restricted to the range 1 to 10 (as in engineering notation for instance) and to bases other than 10 (as in 315 × 2²⁰).

E notation

Most calculators and many computer programs present very large and very small results in scientific notation. Because superscripted exponents like 10⁷ cannot always be conveniently represented on computers, typewriters and calculators, an alternative format is often used: the letter "E" or "e" represents "times ten raised to the power", thus replacing the " × 10ⁿ". The character "e" is not related to the mathematical constant e (a confusion not possible when using capital "E"); and though it stands for exponent, the notation is usually referred to as (scientific) E notation or (scientific) e notation, rather than (scientific) exponential notation (though the latter also occurs).

[edit] Examples

• In the FORTRAN programming language 6.0221415E23 is equivalent to 6.022 141 5 × 10²³.
• The ALGOL 60 programming language uses a subscript ten instead of the letter E, for example 6.02214151023^[1]. ALGOL 68 also allows lowercase E, for example 6.0221415e23.

[edit] Engineering notation

Engineering notation differs from normalized scientific notation in that the exponent b is restricted to multiples of 3. Consequently, the absolute value of a is in the range 1 ≤ |a| <>a| <>

Numbers in this form are easily read out using magnitude prefixes like mega- (b = 6), kilo- (b = 3), milli- (b = −3), micro- (b = −6) or nano- (b = −9). For example, 12.5×10⁻⁹ m can be read as "twelve point five nanometers" or written as 12.5 nm.

[edit] Use of spaces

In normalized scientific notation, in E notation, and in engineering notation, the space (which in typesetting may be represented by a normal width space or a thin space) that is allowed only before and after "×" or in front of "E" or "e" may be omitted, though it is less common to do so before the alphabetical character.^[2]

[edit] Motivation

Scientific notation is a very convenient way to write large or small numbers and do calculations with them. It also quickly conveys two properties of a measurement that are useful to scientists—significant figures and order of magnitude. Writing in scientific notation allows a person to eliminate zeros in front of or behind the significant digits. This is most useful for very large measurements in astronomy or very small measurements in the study of molecules. The examples below display this well.

[edit] Examples

• An electron's mass is about 0.000 000 000 000 000 000 000 000 000 000 910 938 22 kg. In scientific notation, this is written 9.109 382 2 × 10^-31 kg.
• The Earth's mass is about 5,973,600,000,000,000,000,000,000 kg. In scientific notation, this is written 5.9736×10²⁴ kg.
• The Earth's circumference is approximately 40,000,000 m. In scientific notation, this is written 4×10⁷ m. In engineering notation, this is written 40×10⁶ m. In SI writing style, this may be written 40 Mm (40 megameters).
• An inch is 25,400 micrometers. Describing an inch as 2.5400 × 10⁴ µm unambiguously states that this conversion is correct to the nearest micrometer. An approximated value with only 3 significant digits would be 2.54 × 10⁴ µm instead. In this example, the number of significant zeros is actually infinite (which is not the case with most scientific measurements, which have a limited degree of precision). It can be properly written with the minimum number of significant zeros used with other numbers in the application (no need to have more significant digits that other factors or addends). Or a bar can be written over a single zero, indicating that it repeats forever. The bar symbol is just as valid in scientific notation as it is in decimal notation.

[edit] Significant figures

One advantage of scientific notation is that it greatly reduces the ambiguity of number of significant digits. All digits in normalized scientific notation are significant by convention. But in decimal notation any zero or series of zeros next to the decimal point are ambiguous, and may or may not indicate significant figures (when they are they should be underlined to explicitly show that they are significant zeros). In decimal notation, zeros next to the decimal point are not necessarily significant numbers. I.e., they may be there only to show where the decimal point is. In scientific notation, however, this ambiguity is resolved, because any zeros shown are considered significant by convention. For example, using scientific notation, the speed of light in SI units is 2.99792458×10⁸ m/s and the inch is 2.54×10⁻² m; both numbers are exact by definition of the units "inches" per cm and "meters" in terms of the speed of light.^[3] In these cases, all the digits are significant. A single zero or any number of zeros could be added on the right side to show more significant digits, or a single zero with a bar on top could be added to show infinite significant digits (just as in decimal notation).

[edit] Ambiguity of the last digit in scientific notation

It is customary in scientific measurements to record all the significant digits from the measurements, and to guess one additional digit if there is any information at all available to the observer to make a guess. The resulting number is considered more valuable than it would be without that extra digit, and it is considered a significant digit because it contains some information leading to greater precision in measurements and in aggregations of measurements (adding them or multiplying them together).

Additional information about precision can be conveyed through additional notations. In some cases, it may be useful to know how exact the final significant digit is. For instance, the accepted value of the unit of elementary charge can properly be expressed as 1.602176487(40)×10⁻¹⁹ C,^[4] which is shorthand for 1.602176487±40×10⁻¹⁹ °C.

[edit] Order of magnitude

Scientific notation also enables simpler order-of-magnitude comparisons. A proton's mass is 0.000 000 000 000 000 000 000 000 001 672 6 kg. If this is written as 1.6726×10⁻²⁷ kg, it is easier to compare this mass with that of the electron, given above. The order of magnitude of the ratio of the masses can be obtained by comparing the exponents instead of the more error-prone task of counting the leading zeros. In this case, '−27' is larger than '−31' and therefore the proton is roughly four orders of magnitude (about 10,000 times) more massive than the electron.

Scientific notation also avoids misunderstandings due to regional differences in certain quantifiers, such as 'billion', which might indicate either 10⁹ or 10¹².

[edit] Using scientific notation

[edit] Converting

To convert from ordinary decimal notation to scientific notation, move the decimal separator the desired number of places to the left or right, so that the mantissa will be in the desired range (between 1 and 10 for the normalized form). If you moved the decimal point n places to the left then multiply by 10ⁿ; if you moved the decimal point n places to the right then multiply by 10⁻ⁿ. For example, starting with 1,230,000, move the decimal point six places to the left yielding 1.23, and multiply by 10⁶, to give the result 1.23×10⁶. Similarly, starting with 0.000 000 456, move the decimal point seven places to the right yielding 4.56, and multiply by 10⁻⁷, to give the result 4.56×10⁻⁷.

If the decimal separator did not move then the exponent multiplier is logically 10⁰, which is correct since 10⁰ = 1. However, the exponent part "× 10⁰" is normally omitted, so, for example, 1.234 is just written as 1.234 rather than 1.234×10⁰.

To convert from scientific notation to ordinary decimal notation, take the mantissa and move the decimal separator by the number of places indicated by the exponent — left if the exponent is negative, or right if the exponent is positive. Add leading or trailing zeroes as necessary. For example, given 9.5 × 10¹⁰, move the decimal point ten places to the right to yield 95,000,000,000.

Conversion between different scientific notation representations of the same number is achieved by performing opposite operations of multiplication or division by a power of ten on the mantissa and the exponent parts. The decimal separator in the mantissa is shifted n places to the left (or right), corresponding to division (multiplication) by 10ⁿ, and n is added to (subtracted from) the exponent, corresponding to a canceling multiplication (division) by 10ⁿ. For example:

[edit] Basic operations

Given two numbers in scientific notation,

Multiplication and division are performed using the rules for operation with exponential functions:

some examples are:

Addition and subtraction require the numbers to be represented using the same exponential part, so that the mantissas can be simply added or subtracted. These operations may therefore take two steps to perform. First, if needed, convert one number to a representation with the same exponential part as the other. This is usually done with the one with the smaller exponent. In this example, x₁ is rewritten as:

Next, add or subtract the mantissas:

An example:

SCIENTIFIC NOTATION:

During this course you will encounter scientific notation. Below I give a short tutorial of the most important rules. By scientific notation we mean that we express 3,350 as 3.35x10³, 0.00508 as 5.08x10^-3, 1/16 as 6.75x10^-2, etc. (There are special cases, for instance, like numbers between 1 and 100 where we usually do not bother to use scientific notation, e.g., 25 (= 2.5x10¹) and 4 (= 4x10⁰). There are specific rules about adding, subtracting, dividing and multiplying numbers expressed in scientific notation.

ADDITION and SUBTRACTION:
You must make sure that the exponents are the same before you can add or subtract numbers using scientific notation. For example, in the following example the exponents are the same to begin with (=2).
2.4x10² + 1.2x10² = (2.4 + 1.2)x10² = 3.6x10²

In the next example, they have to be made the same.
3.6x10³ - 2.4x10² = 36x10² - 2.4x10² = (36 - 2.4)x10² = 33.6x10² = 3.36x10³.

Further examples:
1.2x10^-2 + 2.4x10^-1 = (0.12 + 2.4)x10^-1 = 2.52x10^-1.
2.3 + 4.2x10² = (0.023 + 4.2)x10² = 4.223x10².

MULTIPLICATION:
You simply multiply the mantissas and add the exponents.
For example:
3.2x10² x 1.5x10³ = (3.2 x 1.5)x10²⁺³ = 4.8x10⁵.
2.6x10^-2 x 2.2x10³ = (2.6 x 2.2)x10^-2+3 = 5.72x10¹ (= 57.2).

DIVISION:
You simply divide the mantissas and subtract the exponents.
For example:
6.6x10³/2.2x10² = (6.6/2.2)x10^3-2 = 3.0x10¹ (= 30).
4.8x10^-2/1.2x10^-3 = (4.8/1.2)x10^-2-(-3) = 4.0x10¹ (= 40).